Finite Group Databasebeta

← C7×#9351 #13186567

Subgroups — C7×#9351 #13186567

Order 1792.

416 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3080C211yesmaximal, characteristic
2C7×#2970C211yesmaximal, characteristic
3C7×#1528C211yesmaximal, characteristic, complemented
4C7×#1488C211yesmaximal, characteristic
5C7×#1470C211yesmaximal, characteristic, complemented
6C7×#1307C211yesmaximal, characteristic, complemented
7C7×#1299C211yesmaximal, characteristic, complemented
8C2×C7×#125C2211yescharacteristic
9C4×C7×#41C2211yescharacteristic
10C7×#479—21no—
11C7×#479—21no—
12C7×#478—21no—
13C7×#478—21no—
14C7×#446—21no—
15C7×#435—22no—
16C7×C8×D4—21no—
17C7×#432—21no—
18C7×#432C2211yescharacteristic
19C7×#432C2211yescharacteristic
20C7×#431C2211yescharacteristic
21C7×#428C411yescharacteristic
22C7×#428C411yescharacteristic
23C7×#423C2211yescharacteristic
24C7×#417C411yescharacteristic
25C7×#417C411yescharacteristic
26C4×C7×M16C2211yescharacteristic
27C2×C7×#97—21no—
28C7×#330—21no—
29C7×#330—21no—
30C7×Q8⋊C8—21no—
31C7×D4⋊C8—21no—
32C7×#324—22no—
33#9351C711yesmaximal, characteristic, direct product factor, Sylow
34C2×C7×#41C2×C411yescharacteristic
35C2×C7×M16—21no—
36C2×C7×M16C2×C411yescharacteristic
37C2×C7×M16C2×C411yescharacteristic
38C22×C7×C8—22no—
39C7×#125—21no—
40C7×#125—21no—
41C7×#125—21no—
42C7×#125—21no—
43C4×C7×Q8D411yescharacteristic
44C4×C7×Q8D411yescharacteristic
45C4×C7×D4—21no—
46C4×C7×D4—21no—
47C4×C7×D4D411yescharacteristic
48C4×C7×D4D411yescharacteristic
49C7×#117—21no—
50C7×#117C2×C411yescharacteristic
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4C2×C411yescharacteristic
54C2×C7×C4⋊C4C2×C411yescharacteristic
55C2×C42×C7—21no—
56C2×C42×C7C2311yescharacteristic
57C7×#107—21no—
58C7×#107—21no—
59C7×#106—21no—
60C7×#106—21no—
61C7×C4⋊C8—22no—
62C7×C4⋊C8—21no—
63C7×C4⋊C8—21no—
64C7×C4⋊C8—21no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×#103—21no—
68C7×#103—21no—
69C7×#102—21no—
70C7×#102—21no—
71C7×C22⋊C8—22no—
72C7×C22⋊C8—21no—
73C7×C22⋊C8—21no—
74C7×C22⋊C8D412yes—
75C7×#97—22no—
76C7×#97—21no—
77C7×#97—21no—
78C7×#97D411yescharacteristic
79C7×#97D411yescharacteristic
80C4×C7×C8—21no—
81C4×C7×C8—21no—
82#3080C2×C711yescharacteristic
83#2970C2×C711yescharacteristic
84#1528C2×C711yescharacteristic, complemented
85#1488C2×C711yescharacteristic
86#1470C2×C711yescharacteristic, complemented
87#1307C2×C711yescharacteristic, complemented
88#1299C2×C711yescharacteristic, complemented
89C7×#41—22no—
90C7×#41—41no—
91C2×C7×Q8C22⋊C411yescharacteristic
92C2×C7×D4—21no—
93C2×C7×D4C22⋊C411yescharacteristic
94C22×C4×C7—21no—
95C22×C4×C7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7C22×C411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7#4111yescharacteristic
102C7×M16—42no—
103C7×M16—41no—
104C7×M16—41no—
105C2×C7×C8—21no—
106C2×C7×C8—21no—
107C2×C7×C8—21no—
108C2×C7×C8—21no—
109C2×C7×C8—22no—
110C2×C7×C8—44no—
111C2×C7×C8—41no—
112C2×C7×C8—41no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—41no—
120C7×C4⋊C4—41no—
121C7×C4⋊C4—41no—
122C7×C4⋊C4—41no—
123C7×C4⋊C4C22⋊C411yescharacteristic
124C7×C4⋊C4C22⋊C411yescharacteristic
125C7×C22⋊C4—21no—
126C7×C22⋊C4—21no—
127C7×C22⋊C4—21no—
128C42×C7—21no—
129C42×C7—21no—
130C42×C7—21no—
131C42×C7C2×D411yescharacteristic
132C42×C7C2×D411yescharacteristic
133C42×C7C2×D411yescharacteristic
134C42×C7#4111yescharacteristic
135C2×#125C22×C711yescharacteristic
136C4×#41C22×C711yescharacteristic
137#479—21no—
138#479—21no—
139#478—21no—
140#478—21no—
141#446—21no—
142#435—22no—
143C8×D4—21no—
144#432—21no—
145#432C22×C711yescharacteristic
146#432C22×C711yescharacteristic
147#431C22×C711yescharacteristic
148#428C4×C711yescharacteristic
149#428C4×C711yescharacteristic
150#423C22×C711yescharacteristic
151#417C4×C711yescharacteristic
152#417C4×C711yescharacteristic
153C4×M16C22×C711yescharacteristic
154C2×#97—21no—
155#330—21no—
156#330—21no—
157Q8⋊C8—21no—
158D4⋊C8—21no—
159#324—22no—
160C23×C7—21no—
161C23×C7C4×D411yescharacteristic
162C7×Q8—21no—
163C7×Q8—21no—
164C7×D4—21no—
165C7×D4—42no—
166C7×D4#10112yes—
167C2×C4×C7—21no—
168C2×C4×C7—21no—
169C2×C4×C7—21no—
170C2×C4×C7—21no—
171C2×C4×C7—21no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—42no—
175C2×C4×C7—42no—
176C2×C4×C7—42no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—81no—
180C2×C4×C7—81no—
181C2×C4×C7C2×C22⋊C411yescharacteristic
182C2×C4×C7C4×D411yescharacteristic
183C2×C4×C7#12011yescharacteristic
184C2×C4×C7#12111yescharacteristic
185C2×C4×C7#12111yescharacteristic
186C2×C4×C7#12311yescharacteristic
187C7×C8—22no—
188C7×C8—42no—
189C7×C8—41no—
190C7×C8—41no—
191C7×C8—41no—
192C7×C8—81no—
193C7×C8—81no—
194C2×#41C2×C4×C711yescharacteristic
195C2×M16—21no—
196C2×M16C2×C4×C711yescharacteristic
197C2×M16C2×C4×C711yescharacteristic
198C22×C8—22no—
199#125—21no—
200#125—21no—
201#125—21no—
202#125—21no—
203C4×Q8C7×D411yescharacteristic
204C4×Q8C7×D411yescharacteristic
205C4×D4—21no—
206C4×D4—21no—
207C4×D4C7×D411yescharacteristic
208C4×D4C7×D411yescharacteristic
209#117—21no—
210#117C2×C4×C711yescharacteristic
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C2×C4×C711yescharacteristic
214C2×C4⋊C4C2×C4×C711yescharacteristic
215C2×C42—21no—
216C2×C42C23×C711yescharacteristic
217#107—21no—
218#107—21no—
219#106—21no—
220#106—21no—
221C4⋊C8—22no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227#103—21no—
228#103—21no—
229#102—21no—
230#102—21no—
231C22⋊C8—22no—
232C22⋊C8—21no—
233C22⋊C8—21no—
234C22⋊C8C7×D412yes—
235#97—22no—
236#97—21no—
237#97—21no—
238#97C7×D411yescharacteristic
239#97C7×D411yescharacteristic
240C4×C8—21no—
241C4×C8—21no—
242C22×C7—21no—
243C22×C7—21no—
244C22×C7—22no—
245C22×C7—21no—
246C22×C7—41no—
247C22×C7—41no—
248C22×C7#38611yescharacteristic
249C4×C7—21no—
250C4×C7—21no—
251C4×C7—21no—
252C4×C7—21no—
253C4×C7—21no—
254C4×C7—41no—
255C4×C7—41no—
256C4×C7—41no—
257C4×C7—81no—
258C4×C7—81no—
259C4×C7C2×#10111yescharacteristic
260C4×C7#41311yescharacteristic
261#41—22no—
262#41—41no—
263C2×Q8C7×C22⋊C411yescharacteristic
264C2×D4—21no—
265C2×D4C7×C22⋊C411yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4C22×C4×C711yescharacteristic
272C22×C4C2×C7×D411yescharacteristic
273C22×C4C7×#4111yescharacteristic
274M16—42no—
275M16—41no—
276M16—41no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—22no—
282C2×C8—44no—
283C2×C8—41no—
284C2×C8—41no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4C7×C22⋊C411yescharacteristic
296C4⋊C4C7×C22⋊C411yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42C2×C7×D411yescharacteristic
304C42C2×C7×D411yescharacteristic
305C42C2×C7×D411yescharacteristic
306C42C7×#4111yescharacteristic
307C2×C7—41no—
308C2×C7—42no—
309C2×C7#188111yescharacteristic
310C2×C7#190111yescharacteristic
311C2×C7#190611yescharacteristic
312C23—21no—
313C23C4×C7×D411yescharacteristic
314Q8—21no—
315Q8—21no—
316D4—21no—
317D4—42no—
318D4C7×#10112yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4C2×C7×C22⋊C411yescharacteristic
334C2×C4C4×C7×D411yescharacteristic
335C2×C4C7×#12011yescharacteristic
336C2×C4C7×#12111yescharacteristic
337C2×C4C7×#12111yescharacteristic
338C2×C4C7×#12311yescharacteristic
339C8—22no—
340C8—42no—
341C8—41no—
342C8—41no—
343C8—41no—
344C8—81no—
345C8—81no—
346C7#935111yescharacteristic, direct product factor, Sylow
347C22—21no—
348C22—21no—
349C22—22no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22C7×#38611yescharacteristic
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—81no—
363C4—81no—
364C4C2×C7×#10111yescharacteristic
365C4C7×#41311yescharacteristic
366C2—41no—
367C2—42no—
368C2C7×#188111yescharacteristic
369C2C7×#190111yescharacteristic
370C2C7×#190611yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC2×C7×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C7×#1528C2semidirect
5. C7×#1470C2semidirect
6. C7×#1307C2semidirect
7. C7×#1299C2semidirect
33. #9351C7direct (N × H)
84. #1528C2×C7semidirect
86. #1470C2×C7semidirect
87. #1307C2×C7semidirect
88. #1299C2×C7semidirect
346. C7#9351direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C7×#30802
C7×#29702
C7×#15282
C7×#14882
C7×#14702
C7×#13072
C7×#12992
#93517